Maharashtra Board Class 12 Maths Commerce Part I Chapter 2 Matrices PDF Download

Matrices

Let's Study

Types of Matrices

Algebra of Matrices

Properties of Matrices

Elementary Transformation

Inverse of Matrix

Application of Matrices

Determinant of a Matrix

Let's Recall

Determinant of a Matrix

2.1 Introduction

The theory of matrices was developed by the mathematician Arthur Cayley. Matrices are useful in expressing numerical information in a compact form. They are effectively used in expressing different operations. Hence they are essential in economics, finance, business and statistics.

Definition: A rectangular arrangement of mn numbers in m rows and n columns, enclosed in [ ] or ( ) is called a matrix of order m by n. A matrix by itself does not have a value or any special meaning.

The order of a matrix is denoted by m × n, read as m by n.

Each member of a matrix is called an element of the matrix.

Matrices are generally denoted by capital letters like A, B, C, …. and their elements are denoted by small letters like aij, bij, cij, ….. etc. where aij is the element in ith row and jth column of the matrix A.

For example: i) A = \[\begin{bmatrix} 2 & 3 & 9 \\ 1 & 0 & -7 \\ 4 & -2 & 1 \end{bmatrix}\] here a₃₂ = −2

A is a matrix having 3 rows and 3 columns. The order of A is 3×3. There are 9 elements in the matrix A.

ii) B = \[\begin{bmatrix} 1 & 5 \\ 2 & 0 \\ 6 & 9 \end{bmatrix}\]

B is a matrix having 3 rows and 2 columns. The order of B is 3×2. There are 6 elements in the matrix B.

In general, a matrix of order m × n is represented by

Teacher's Note

Matrices are used in real life. Banks use matrices to store data about customers. Schools use matrices to organize student information.

Exam Trick

Remember: Order means rows × columns. A 3×2 matrix has 3 rows and 2 columns. Count carefully on exam day.

Points to Remember

A matrix is a rectangular arrangement of numbers in rows and columns.

Order of a matrix is m × n where m is rows and n is columns.

Element aij means element in ith row and jth column.

Matrices are shown with capital letters like A, B, C.

Matrix elements are shown with small letters like aij.

2.2 Types of Matrices

1) Row Matrix: A matrix that has only one row is called a row matrix. It is of order 1 × n, where n ≥ 1.

For example: i) \([1 \quad 2 \quad -1 \quad 2]\) ii) \([9 \quad 0 \quad 3 \quad 1 \quad 3]\)

2) Column Matrix: A matrix that has only one column is called a column matrix. It is of order m × 1, where m ≥ 1.

For example: i) \(\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}\) ii) \(\begin{bmatrix} 5 \\ 9 \\ 3 \end{bmatrix}\)

Note: A real number can be treated as a matrix of order 1×1. It is called a singleton matrix.

3) Zero or Null Matrix: A matrix in which every element is zero is called a zero or null matrix. It is denoted by O.

For example: O = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}\)

4) Square Matrix: A matrix with the number of rows equal to the number of columns is called a square matrix. If a square matrix is of order n×n then n is called the order of the square matrix.

For example: A = \(\begin{bmatrix} 5 & 3 & i \\ 1 & 0 & -7 \\ 2 & 8 & 9 \end{bmatrix}\)

Let's Note: Let A = [aij]n×n be a square matrix of order n, Then

(i) the elements a₁₁, a₂₂, a₃₃, …., aii, …., ann are called the diagonal elements of the matrix A. Note that the diagonal elements are defined only for a square matrix.

(ii) elements aij where i ≠ j are called non-diagonal elements of the matrix A.

Teacher's Note

A square matrix has equal rows and columns. A 3×3 matrix is square, but 3×2 is not square. This is like a perfect square number.

Exam Trick

Remember: Square matrix = rows = columns. Just like 3×3 is a perfect square, a 3×3 matrix is a square matrix.

Points to Remember

Row matrix has only one row.

Column matrix has only one column.

Square matrix has rows = columns.

Diagonal elements are a₁₁, a₂₂, a₃₃ and so on.

Zero matrix has all elements equal to zero.

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